Heun

Heun's (aka improved Euler) time integration method.

Heun's (or "improved Euler") time integration method is second-order accurate in time. It is a two-step explicit method and a special case of the 2nd-order Runge-Kutta method.

Description

With $var element = document.getElementById("moose-equation-0fff5960-b4b6-4a75-a338-a3edf4833d3a");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-28658ccc-f29e-4aa2-8b38-c261846ef6b9");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-9b9fe653-3faa-4480-bbc7-db763f5bb6fc");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-802497d7-b5dc-4096-9e16-ae6fafccfd9e");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-3cd470aa-dc98-4e8a-8ade-26ed90003743");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, Heun's method can be written:

var element = document.getElementById("moose-equation-a6f84056-8c1e-4938-bfb8-d29ac2e458a8");katex.render("U(t+\\Delta t) = U(t) + \\dfrac{\\Delta t}{2} \\left(A(t, U(t)) + A(t,U(t) + \\Delta t A(t, U(t)) ) \\right)", element, {displayMode:true,throwOnError:false});

This method can be expressed as a Runge-Kutta method with the following Butcher Tableau:

var element = document.getElementById("moose-equation-685fc69d-c809-4690-9214-3736708c00fd");katex.render("\\begin{array}{c|cc} 0 & 0 \\\\ 1 & 1 & 0 \\\\ \\hline & 1/2 & 1/2 \\end{array}", element, {displayMode:true,throwOnError:false});

warning

All kernels except time-(derivative)-kernels should have the parameter implicit=false to use this time integrator.

warning

ExplicitRK2-derived TimeIntegrators (ExplicitMidpoint, Heun, Ralston) and other multistage TimeIntegrators are known not to work with Materials/AuxKernels that accumulate 'state' and should be used with caution.

Input Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.